Remote Sensing of Water and Environment

Chapter 7: Microwave Radiometry and Radiative Transfer

Ardeshir Ebtehaj
University of Minnesota
Table of Contents
The bulk of the energy recevied by the Earth is in the form of solar EM radiation. Part of this incident radiation is scattered or absorbed by the Earth's atmosphere and the remainder is transmitted to the Earth's surface. Of the EM waves that reach the surface, some will be scattered outward and the rest will be absorbed. Absorbed EM energy is transformed into heat.
The transformation process of EM energy across the Earth's materials is explained by the theory of radiative transfer.
Radiometry is the field of science and engineering concerned with the measurement of EM radiation.
The term radiometry means the measurement of incoherent EM energy. All materials emit (radiate) EM energy.

7-1- Planck's Blackbody Radiation Law

A blackbody radiates uniformly in all directions with a spectral brightness intensity,
where
Over a narrow frequency interval centered at f, the brightness intensity is given by
It is sometimes of interest to express the brightness intensity in terms of wavelength λ. Knowing that
Image 6.3.png
Planck's radiation law [adopted from Kraus, 1966]
Image 6.4.png
Geometry for power received from a blackbody source
Let's asume that is the power [watts] emitted by a blackbody source area of , through a solid angle of , and over a bandwidth of . The blackbody source has an area of . Our objective is to determine spectral power intercepted by a receiver with an aperture area at at a range R from the source.
[]
And thus,
For a receiver with a bandwidth
.

7-2- The Rayleigh-Jeans Law

The dashed line in the following figure provides an excellent approximation to Planck's law, as long as the frequency f is well below , where is the maximum. The law that explains this dashed line is known as the Rayleigh-Jeans law.
Figure 2-1: Comparison of Planck's law with its low-frequency approximation (Rayleigh-Jeans law) at 300 K.
Note:
Thus, if in Planck's law, the expression simplifies to
Eq. (7.1)
The law is valid when or . Above 300 GHz, the deviation from the Planck's law is around 3%.

7-3- Antenna Pattern

Each specific combination of the zenith angle and the azimuth angle denotes a specific direction in a spherical coordinate system.
The directional pattern of an antenna is described by its normalized radiation intensity , defined as the ratio of the power density at a specified range R to , the maximum value of at the same range,
(dimensionaless).
The normalized radiation intensity characterizes the directional pattern of the energy radiated or recieved by the antenna.
Some antennas exhibit highly directive patterns with narrow beams, in which case it is often convenient to plot the antenna pattern on a decibel scale by expressing in decibels:
Image 6.6.png
Blackbody spectral brightness incident antenna with effective aperture and radiation pattern .
Representative plots of the normalized radiation pattern of a microwave antenna in (left) polar form and (right) rectangular form.
For an antenna with a single mainlobe, the pattern solid angle describes the equivalent width of the mainlobe of the antenna pattern as shown in the following figure.
The pattern solid angle defines an equivalent cone over which all the radiation of the actual antenna is concentrated with uniform intensity equal to the maximum of the actual pattern.

7-4- Power and Temperature Correspondence

Consider the above arrangement for a lossless receiver antenna with effective aperture , surrounded by a blackbody of spectral intensity . The antenna is characterized by a radiation pattern . Therefore
The total power received is
Emission by a blackbody is unpolarized. Therefore, assume half of the energy is in vertical and half is in horizontal polarization.
Using the Rayleigh-Jeans approximation, we have
If the detected power is limited to a narrow bandwidth, .
Solving the integral using:
,
leads to the final equation
This result is of fundamental significance in microwave remote sensing. The direct linear relationship between power and temperature lends to the interchangeable use of them. Note thet is radiation power by a balck body.

7-5- Radiation by Natural Materials

7-5-1- Brightness Temperature

A blackbody is an idealized body and a perfect absorber.
Real materials, sometimes referred to as gray body, emit less than a blackbody and do not necessarily absorb all the incident energy upon them.
In the microwave region, given Eq. (6.2.1), the unpolarized brightness intensity of a blackbody at temperature T is
For a gray body, the directional emission/absorption, where the Rayleigh-Jeans law is valid in microwave bands, is as follows:
where is the brightness temperature [K]. The emissivity, is expressed as
---------------------------------------------------------IMPORTANT
Image 6.7.png
Brightness temperature of a semi-infinite isothermal medium.
Since , it follows that . Thus, . It is customary to use instead of , especially in microwave region.
---------------------------------------------------------IMPORTANT

7-6- Distribution of Brightness Temperature

The radiation incident upon the antena from any specific direction may contain components originating from several different sources.
  1. Upward surface emission -
  2. Upward atmospheric emission -
  3. Downward atmospheric emission - that gets scattered/reflected by the terrain (surface scattering) in the direction of antenna
Image 6.8a.png Image 6.8b.png
Relationship between (lossless) antenna temperature , brightness temperature incident on the antenna and surface emitted brightness temperature : a) schematic representation; b) block-diagram representation.
The net brightness temperature , representing the energy incident upon the antenna, is given by
Examples of configurations of interest in radiometric remote sensing.

7-6- Theory of Radiative Transfer

The purpose here is to develop formulations to relate observed brightness temperatures incident upon radiometer antenna aperture to the physical and electromagnetic properties of the scene.
Interaction between electromagnetic radiation and matters involves two simultaneous processes:
  1. Extinction: radiation tranversing a medium is reduced in intenity due to absorption and/or scattering by the medium.
  2. Emission: when the medium adds energy of its own to the original beam of energy.
Image 6.10.png
Radiation transfer through an infinitesimal cylinder

7-6-1 Equation of Radiative Transfer

Brightness intensity is energy per unit area dA, radiated within a solid angle , during an interval of with a frequency bandwidth . The loss in intensity due to propogation over thickness dR is
(Eq. 7.2)
where is the extinction coefficient . The extinction coefficient can be expressed as
where and are obsorption and scattering coefficients.
In addition to alternating the incident brightness temperature, the medium adds its own emission along :
where and are source functions:
The scattering source function accounts for radiation scattered in the direction due to radiation incident upon the cylinder from all other directions.
where is the brightness intensity incident from the direction , a portion of which gets redirected along the direction , thereby adding to the original brightness intensity.
Scattering is not necessarily an isotropic process due to the shape, size, orientation and spatial distribution of particles that scatter. This heterogeneity is accounted for by , or the scattering phase function. In an isotropic medium,
Oftentimes, and are expressed as
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(single scattering albedo)
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(Eq. 7.3)
where is called a total source function. Propogation over thickness dR results in a change in intensity dI given by
The dimensionless is called the optical or electromagnetic depth.
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Therefore, the radiative tranfer equation is
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7-6-2 Theory of Radiative Transfer for Brightness Temperature

Rayleigh-Jeans law:
Absorption source function:
Scattering source function:
where is the volume scattering radiometric temperature.
Therefore, given that , the radiative transfer equation for brightness temperature is
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Eq. (7.4)
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7-6-3 Brightness Temperature of a Stratified Medium

Image 6.11.png
Geometry for brightness temperature radiative transfer equation
Let us assume that propogation properties only change as a function of z. We want to solve the radiative transfer equation to obtain .
Let us multiple both sides of the radiative transfer equation for , Eq. (7.4.), by
where . The left handed side can be written as
(Eq. 7.5)
In the last step we used for
which can be proven by
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At a height , Eq. (7.5) becomes
where is the upward emission contribution
(Eq. 7.6)
and is the surface emission + reflected downward atmospheric or vegetation emission.
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Note that gets attenuated from to along its traveling path. The one-way atomspheric transmissivity between and is
.
.

7-6-4 Brightness Temperature of a Scatter-Free Medium

In a scatter free medium we can assume . Therfore, Eq. (7.5) reduces to
(optical depth)
For a constant and temperature, we can reduce the above equation to the following
and
Volume Scattering
Snow particles range from 0.1-5 mm. If the wavelength is much larger than the grain size, the medium appead electromagnetically homogeneous with no appreciable scattering. If the wavelength is on the same order of magnitude as particle size, scattering will take place.
Scatter-free
Scattering will take place

7-6-5 Upwelling and Downwelling Atmospheric

In the absence of scattering, the upwelling atmospheric brightness temperature is
.
Models for and are available for Earth's atmosphere.
Image 6.12.png
Upward and downward emission contributions of a plane-stratified atmosphere
A ground-based upward-looking radiometer observes a downwelling atmospheric brightness temperature given by
The physical interpretation of the above equation is quite simple; the energy emitted by a stratum at height and of vertical thickness (slant thickness ) is proportional to , which, after propagation down to the surface, is reduced by the factor due to absorption by the intervening layers.
For a planar, homogeneous atmosphere with , , and depth H,
Note that where is the zenith optical thickness of the atmospheric layer
so for a very optically thick atmosphere , Then we have

7-7- Terrain Brightness Temperature

Image 6.13a.png Image 6.13b.png Image 6.13c.png
Three scenarios for medium 2 and its surface boundary.

7-7-1 Transmission Across a Specular Boundary

Brightness transmission across a planar boundary
The power of the EM wave, for example in horizontal polarization, is
Base on the the Snell's Law, we have
Differentiating the law, one can have
Multiplying both sides with the azimuth angle, we have
Multiplying both sides of the above equation by the Snell's law, one can get
we know that and thus the above equation simplifies to
Given that
, ,
where are the brightness intensities for a grey body medium. Therefore,
or
where the trasnissivity when EM waves travel from medium 2 to medium 1 is, in which are the Fresnel reflectivity for incidence angles.
Similarly for the vertical polarization, we have
where

7-7-2 Emission by a Specular Surface

Medium 2 has a temperature-depth profile
We already described that the slanted emission by a medium can be expleained by the following equation.
For a short interval over which is constant
(note that ).
We have shown that .
If and constitutive parameters of medium 2 are known, we can solve the above equation numerically. For a homogeneous medium with a uniform temperautre , the emission simplifies to .
Therefore, observed brightness temperature by a downlooking radiometer is
where we know that .
Note: is temperature of uppermost layer that represents bulk of the contributions to .
Calculated reflectiveness and emissivities of a specular surface for h and v polarizations at 10 GHz

7-7-3 Emission by a Rough Surface

Specular- and rough-surface sacattering and emission
Generally speaking, increasing surface roughness increases surface emssion radiation than that of a specular surface with the same dielectic constant.
There are semi empirical models for explaining total reflectivity . The basic structure of ,
where is an effective rough surface reflectivity given by
where and are Fresnel reflectivity at polarization p, Q is a polarization mixing factor, is an equivalent root mean squared (rms) height and n is an angular exponent, which is generally polarization dependent.
Parameters derived from regression analyses (Wang et al., 1983)
where : electromagnetic roughness
Angular patterns of the emissivity measured at 1.4 GHz for three bare-soil fields with different surface roughness [Newton and Rouse, 1980]

Two configurations of height variations
For a random surface whose mean is coincide with x-y plane, its height above x-y plane usually is characterized by a Gaussian density :
, where s is the rms height
where and .
The correlation function is defined as
.
The correlation length l is the value of ξ at which .
IMPORTANT: A surface is radiometrically smooth when .
Random, isotropic surface : (a) pictorial view, (b) measured height profile , (c) pdf of digitized height profile, and (d) autocorrelation function where ζ is the displacement between two points on the surface